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Here is my problem:
I have to find the best position of a point respecting some constraint.
The point should be "above" N plane.
So it should respect theses inequation:
a1x + b1y +c1z + d1 > 0
a2x + b2y +c2z + d2 > 0
a3x + b3y +c3z + d3 > 0
.
.
.
.
anx + bny + cnz + dn > 0
Thats the first constraint.
The plane are always defining an existing volume, but this volume is not necessary closed.
The second constraint is that the solution point have to be the closest point to another out of the volume.
ie: the distance between my reference point and the solution point should be the smaller possible.
I have to solve this system numerically.
Please heeelp ;)
PS: I know... I need to improve my english level ;)

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it looks to me that the point the closest to a reference point outside the volume is a point on the surface. So you need to find the closest distance to the n planes, there intersecting edges or vertices.

It sounds like the shortes distance of a point to a polyhedron.

Is this what you mean?

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No... that not my problem,in fact, i want to make a LOD that always enclose the volume of the original mesh.So at the edge collapse, to be sure that the new faces created are enclosing the old faces, a suffisant condition is that the point lie outside the planes defined by the old faces.

So the solution point have to be above these plane.But i also need that the new volume created by the edge collapsing be the smaller possible. So i ve decide to introduce the second constraint (the solution point need to be the closest point respecting the plane inequalities with one of the edge point).

It should work because the mesh is always closed and convex.(the first step of the algo is to build a convex hull of the mesh)

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I want to implement the progressive hulls describe herehttp://research.microsoft.com/~hoppe/silclip.pdfbut i have replace the winding number by the distance between my point solution and one of the point of the edge (coz my mesh is convex)

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Take a look at this picture:http://etudiant.epita.fr/~godin_g/explication.jpg

In this picture u can see that the shortest distance is 0 (plane 1) but the solution doesn't lies on the plane 1.

The picture show the problem in 2D... I'll make another to show my problem in 3d.

to grhodes_at_work:My appliction is a motor3d. I want to use this special lod to create a bounding mesh (approximatly ~30/50 triangles). These bounding mesh will be used for occlusion culling/collision purposes.Im actually using the garland and heckbert algorithme for creating LOD.I want to replace the cost function and the way to find the target vertex. So at each contraction the new triangles will allways enclose the old triangles.To be sure that the new faces are enclosing the old triangles, the condition is that the target vertex be above the plane of all the old triangle.But we also need to minimize the volume created by the contraction. So the target vertex must be above all plane AND be the closest to a reference point.I choose the reference point to be either one point of the edge either the midpoint of the edge.